Represents an
empirical probability distribution -- a probability distribution derived
from observed data without making any assumptions about the functional form
of the population distribution that the data come from.

The JDK provided generator is a simple one that can be used only for very simple needs.
The Mersenne Twister is a fast generator with very good properties well suited for
Monte-Carlo simulation. It is equidistributed for generating vectors up to dimension 623
and has a huge period: 219937 - 1 (which is a Mersenne prime). This generator
is described in a paper by Makoto Matsumoto and Takuji Nishimura in 1998: Mersenne Twister:
A 623-Dimensionally Equidistributed Uniform Pseudo-Random Number Generator, ACM
Transactions on Modeling and Computer Simulation, Vol. 8, No. 1, January 1998, pp 3--30.
The WELL generators are a family of generators with period ranging from 2512 - 1
to 244497 - 1 (this last one is also a Mersenne prime) with even better properties
than Mersenne Twister. These generators are described in a paper by François Panneton,
Pierre L'Ecuyer and Makoto Matsumoto Improved Long-Period
Generators Based on Linear Recurrences Modulo 2 ACM Transactions on Mathematical Software,
32, 1 (2006). The errata for the paper are in wellrng-errata.txt.

For simple sampling, any of these generators is sufficient. For Monte-Carlo simulations the
JDK generator does not have any of the good mathematical properties of the other generators,
so it should be avoided. The Mersenne twister and WELL generators have equidistribution properties
proven according to their bits pool size which is directly linked to their period (all of them
have maximal period, i.e. a generator with size n pool has a period 2n-1). They also
have equidistribution properties for 32 bits blocks up to s/32 dimension where s is their pool size.
So WELL19937c for exemple is equidistributed up to dimension 623 (19937/32). This means a Monte-Carlo
simulation generating a vector of n variables at each iteration has some guarantees on the properties
of the vector as long as its dimension does not exceed the limit. However, since we use bits from two
successive 32 bits generated integers to create one double, this limit is smaller when the variables are
of type double. so for Monte-Carlo simulation where less the 16 doubles are generated at each round,
WELL1024 may be sufficient. If a larger number of doubles are needed a generator with a larger pool
would be useful.

The WELL generators are more modern then MersenneTwister (the paper describing than has been published
in 2006 instead of 1998) and fix some of its (few) drawbacks. If initialization array contains many
zero bits, MersenneTwister may take a very long time (several hundreds of thousands of iterations to
reach a steady state with a balanced number of zero and one in its bits pool). So the WELL generators
are better to escape zeroland as explained by the WELL generators creators. The Well19937a and
Well44497a generator are not maximally equidistributed (i.e. there are some dimensions or bits blocks
size for which they are not equidistributed). The Well512a, Well1024a, Well19937c and Well44497b are
maximally equidistributed for blocks size up to 32 bits (they should behave correctly also for double
based on more than 32 bits blocks, but equidistribution is not proven at these blocks sizes).

The MersenneTwister generator uses a 624 elements integer array, so it consumes less than 2.5 kilobytes.
The WELL generators use 6 integer arrays with a size equal to the pool size, so for example the
WELL44497b generator uses about 33 kilobytes. This may be important if a very large number of
generator instances were used at the same time.

All generators are quite fast. As an example, here are some comparisons, obtained on a 64 bits JVM on a
linux computer with a 2008 processor (AMD phenom Quad 9550 at 2.2 GHz). The generation rate for
MersenneTwister was about 27 millions doubles per second (remember we generate two 32 bits integers for
each double). Generation rates for other PRNG, relative to MersenneTwister:

So for most simulation problems, the better generators like Well19937c and Well44497b are probably very good choices.

Note that none of these generators are suitable for cryptography. They are devoted
to simulation, and to generate very long series with strong properties on the series as a whole
(equidistribution, no correlation ...). They do not attempt to create small series but with
very strong properties of unpredictability as needed in cryptography.